BUS601 Statistics Project

Introduction

The objective of this paper is to provide an analysis of a dataset pertaining to the analysis of the speed of light, which has been a topic of major interest for physicist and scientists in general. During centuries, a myriad of theories were elaborated, some postulating that the speed of light was finite. By the end of 19 ^th century it became clear that light traveled at a finite constant speed, and multiple experiments were carried out to determine what the speed of light is. Michelson made 100 determinations of the velocity of light in air, and the resulting velocities obtained are included at the StatLib depository of Carnegie Mellon University at http://lib.stat.cmu.edu/DASL/Datafiles/Michelson.html . Nowadays, we know that the actual speed of light is 299,792.5 km/sec. Our main research question in this paper is to statistically assess whether or not Michelson measurements differ from what we know today as the speed of light.

Charts

The following histogram is obtained from the dataset described above:

The distribution appears to be fairly normal (In fact, the distribution doesn’t depart significantly from normality, p = 0.255. See the appendix)

Descriptive Statistics

The following is obtained

We are interested in testing the following null and alternative hypotheses

\[\begin{array}{*{35}{l}}

{{H}_{0}}:\mu =299,792.5\,\,\,\text{km/sec} \\

{{H}_{A}}:\mu \ne 299,792.5\,\,\,\text{km/sec} \\

Considering that the population standard deviation is not known, we have to use a t-test using the following expression:

This corresponds to a two-tailed t-test. The t-statistics is computed as follows:

The two-tailed p-value for this test is computed as

Considering that the p-value is such that , and this means that we reject the null hypothesis H ₀ .

Therefore, we have enough evidence to support the claim that the mean differs significantly from the true speed of light.

We also get that the 95% confidence interval is found to be

\[CI=\left( 299852.4-1.984\times \frac{79.011}{\sqrt{100}},\,\,299852.4+1.984\times \frac{79.011}{\sqrt{100}} \right)=\left( 299,836.72,\,\,299,868.08 \right)\]

This means that we are 95% confident that the true population mean \(\mu\) is contained by the interval \(\left( 299,836.72\text{ km/sec},\,\text{ }299,868.08\text{ km/sec} \right)\).

Confidence Interval for the Standard Deviation

The 95% confidence interval for the population standard deviation is computed as

\[CI=\left( \sqrt{\frac{\left( n-1 \right){{s}^{2}}}{\chi _{U}^{2}}},\,\,\sqrt{\frac{\left( n-1 \right){{s}^{2}}}{\chi _{L}^{2}}}\, \right)\]

where \(\chi _{U}^{2}\) and \(\chi _{L}^{2}\) represent the upper and lower critical Chi-Square values. In this case, we have \(\alpha =0.05\) and \(n-1 = 100-1 = 99\) degrees of freedom. These critical values are computed as

\[\chi _{U}^{2}=128.422,\,\,\,\chi _{L}^{2}=73.361\]

The confidence interval for the population standard deviation is computed as

\[CI=\left( \sqrt{\frac{\left( n-1 \right){{s}^{2}}}{\chi _{U}^{2}}},\,\,\sqrt{\frac{\left( n-1 \right){{s}^{2}}}{\chi _{L}^{2}}}\, \right) = \left( \sqrt{ \frac{\left( 100-1 \right)\times {79.011}^2}{128.422} },\,\,\,\sqrt{ \frac{\left( 100-1 \right)\times {79.011}^2}{73.361} } \right)\]

\[=\left( 69.372\text{ km/sec},\,\,91.785\text{ km/sec} \right)\]

The 95% confidence interval for the population variance corresponds to

\[CI=\left( 4812.504,\,\,8424.506 \right)\]

This implies that we are 95% confident that the actual population standard deviation \(\sigma\) is contained by the interval \(\left( 69.372,\,\text{ }91.785 \right)\).

Conclusions

The sample data shows that Michelson was fairly right on target on his measurements of speed of light, and in fact, his measurements don’t differ significantly from the true speed of light, at the 0.05 significance level. Also, Michelson experiment must have been very thoughtfully designed, because the measurements exhibit a very low variability, with a 95% confidence interval for the standard deviation of (69.372 km/sec, 91.785 km/sec)

Appendix

Normality Test

Price: $10.66

Solution: The downloadable solution consists of 6 pages, 466 words and 8 charts.
Deliverable: Word Document